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LAYERING AS OPTIMIZATION DECOMPOSITION:
QUESTIONS And ANSWERS
Mung Chiang, Steven H. Low, A. Robert Calderbank, John C. Doyle
Abstract— Network protocols in layered architectures
have historically been obtained on an ad-hoc basis, and
much of the recent cross-layer designs are conducted
through piecemeal approaches. Network protocols may instead be holistically analyzed and systematically designed
as distributed solutions to some global optimization problems in the form of generalized Network Utility Maximization (NUM), providing insight on what they optimize and on
the structures of network protocol stacks.
In the form of 10 Questions and Answers, this paper
presents a short survey of the recent efforts towards a systematic understanding of “layering” as “optimization decomposition”. The overall communication network is modeled by a generalized NUM problem, each layer corresponds to a decomposed subproblem, and the interfaces
among layers are quantified as functions of the optimization variables coordinating the subproblems. Furthermore,
there are many alternative decompositions, each leading
to a different layering architecture. Industry adoption of
this unifying framework has also started. Here we summarize the current status of horizontal decomposition into distributed computation and vertical decomposition into functional modules such as congestion control, routing, scheduling, random access, power control, and coding. We also discuss under-explored future research directions in this area.
More importantly than proposing any particular crosslayer design, this framework is working towards a mathematical foundation of network architectures and the design
process of modularization.
Keywords: Adaptive coding, Cross-layer design, Congestion control, Distributed algorithm, Lagrange duality,
MAC, Network architecture, Network utility maximization, Optimization, Power control, Reverse engineering,
Routing, TCP/IP, Scheduling, Stochastic control, Wireless
ad hoc networks.
I. OVERVIEW
In order to practically realize the potential of networkcentric technology, system-level analysis and design for
M. Chiang is with Electrical Engineering Department, Princeton University, [email protected].
S. H. Low is with
Computer Science and Electrical Engineering Departments, Caltech, [email protected].
A. R. Calderbank is with Electrical
Engineering and Mathematics Departments, Princeton University,
[email protected]. J. C. Doyle is with Control and Dynamic
Systems Department, Caltech, [email protected].
performance and robustness must become far more systematic than the state-of-the-art, and ideally should be
based on a firm theoretical foundation and first principles.
While the individual subjects of communications, control,
and computation have rich theories of practical relevance,
they are fragmented and incompatible. Network-level design must coherently integrate all of these issues and until
recently has been largely ad hoc and heuristic. This is
inadequate both for the evolution of existing network infrastructure and the successful deployment of future, advanced network-centric technologies. We argue that the
viewpoint of “architecture first” is essential and outline
the progress and promise of a new framework of “layering as optimization decomposition”.
The idea that network architectures may be mathematically understood is a bold one. Network architecture determines functionality allocation: “who does what” and
“how to connect them”, rather than just resource allocation. Layered architectures form one of the most fundamental and influential structures of network design. It
adopts a modularized and often distributed solution approach to network coordination and resource allocation.
Each layer controls a subset of the decision variables, and
observes a subset of constant parameters and the variables
from other layers. Intuitively, layered architectures enable
a scalable, evolvable, and implementable network design
while introducing potential risks to manageability of the
network. There is clearly more than one way to “divide
and conquer” the network design problem. For example,
from a data-plane performance point of view, some layering schemes may be more efficient or more fair than others. Examining the choices of modularized design for networks, we would like to tackle the question of “how to”
and “how not to” layer. Most of the recent work focus on
resource allocation functionalities and performance metrics only. The limitations of such focus will also be discussed here.
Each layer in the protocol stack hides the complexity of
the layer below and provides a service to the layer above.
While the general principle of layering is widely recognized as one of the key reasons for the enormous success
of data networks, there is little quantitative understanding to guide a systematic, rather than an ad hoc, process
of designing layered protocol stack for wired and wireless networks. One possible perspective to rigorously and
holistically understand layering is to integrate the various
protocol layers into a single coherent theory, by regarding
them as carrying out an asynchronous distributed computation over the network to implicitly solve a global optimization problem. Different layers iterate on different
subsets of the decision variables using local information
to achieve individual optimality. Taken together, these local algorithms attempt to achieve a global objective. Such
a design process of modularization can be quantitatively
understood through the mathematical language of decomposition theory for constrained optimization [51].
Such a framework of “layering as optimization decomposition” exposes the interconnection between protocol
layers and can be used to study rigorously the performance tradeoff in protocol layering, as different ways
to modularize and distribute a centralized computation.
Even though the design of a complex system will always
be broken down into simpler modules, this theory will allow us to systematically carry out this layering process
and explicitly trade off design objectives.
The key idea in “layering as optimization decomposition” is as follows. Different vertical decompositions of
an optimization problem, in the form of a generalized Network Utility Maximization (NUM), are mapped to different layering schemes in a communication network, with
each decomposed subproblem in a given decomposition
scheme corresponds to a layer, and functions of primal or
Lagrange dual variables (coordinating the subproblems)
correspond to the interfaces among the layers. Horizontal
decompositions can be further carried out within one functionality module into distributed computation and control
over geographically disparate network elements. Since
different decompositions lead to alternative layering architectures, we can also tackle the question “how and how
not to layer” by investigating the pros and cons of decomposition techniques. Furthermore, by comparing the
objective function values under various forms of optimal
decompositions and suboptimal decompositions, we can
seek “separation theorems” among layers: conditions under which layering incurs no loss of optimality. Robustness of these separation theorems can be further characterized by sensitivity analysis in optimization theory: how
much will the differences in the objective value (between
different layering schemes) fluctuate as constant parameters in the generalized NUM formulation are perturbed.
There have been many recent research activities along
the above direction by research groups around the world.
In Section II, we use an innovative format in this paper
to explain the “common language” of “layering as op-
timization decomposition”. We pose 10 skeptical questions that critically challenge the meaning and utility of
the framework, and present the associated answers as we
understand them today. Some of the answers are far from
being complete, reflecting the fact this research area is a
very active one that still needs much future work. Some
key messages are summarized in Section III.
This short paper only provides a high-level tutorial of
the collective efforts by many in the research community
in this area. A more comprehensive survey can be found
in [10].
II. T EN Q UESTIONS AND A NSWERS
A. Isn’t this just another cross-layer design? What kind
of architectural issues does it resolve?
Answer: There are two intellectually fresh cornerstones
behind “layering as optimization decomposition”. The
first is “network as an optimizer”. The idea of “protocol as a distributed solution” to some global optimization
problem (in the form of the basic NUM) has been successfully tested in trials for Transmission Control Protocol (TCP) [26]. The key innovation from this line of work
[29], [34], [35], [44], [45], [47], [55] is to view TCP/IP
network as an optimization solver and each variant of congestion control protocol as a distributed algorithm solving
a specified basic NUM with a different utility function. In
the basic NUM, the objective is the sum of source utilities as functions of rates, the constraints are linear flow
constraints, and optimization variables are source rates.
Other recent results also show how to reverse engineer
Border Gateway Protocols (BGP) as solving the Stable
Path Problem [20], and contention-based Medium Access
Control (MAC) protocols as game-theoretic selfish utility
maximization [38], [59]. Starting from a given protocol
originally designed based on engineering heuristics, reverse engineering discovers the underlying mathematical
problems being solved by the protocols and demonstrates
the application of derived insights through forward engineering improvements of the protocols.
The second key concept is “layering as decomposition”.
As will be discussed in the answers to the next two questions, generalized NUM problems can be formulated to
represent a network design problem. These generalized
NUM problems put the end user utilities at the “driver’s
seat” for network design. For example, benefits of innovations in physical layers through better modulation and
coding schemes are now characterized by the enhancement to applications rather than just the drop in bit error
rates, which the users do not directly observe. An optimal
solution to a generalized NUM formulation automatically
establishes the benchmark for all layering schemes. The
problem itself does not have any pre-determined layering
architecture. Indeed, layering is a human engineering effort.
The overarching question then becomes how to attain
an optimal solution to a generalized NUM in a modularized and distributed way. Vertical decompositions
across modules and horizontal decompositions across disparate network elements can be conducted systematically
through the theory of decomposition for nonlinear optimization. Implicit message passing (where the messages
have physical meanings and need to be measured anyway)
or explicit message passing quantifies the amount of information sharing and decision coupling required for a particular decomposition.
There are many different ways to decompose a given
problem, each of which corresponds to a different layering architecture. Even a different representation of the
same NUM problem can lead to different decomposability structures even though the optimal solution remains the
same. These decompositions, i.e., layering schemes, have
different characteristics in efficiency, robustness, asymmetry of information and control, and tradeoff between
computation and communication. Some are “better” than
others depending on the criteria set by the network users
and managers. A systematic exploration in the space of
alternative decompositions is possible, where each particular decomposition represents a holistically designed protocol stack.
Given the layers, crossing layers is tempting. For example, layers can be crossed for wired or wireless networks
in many different ways. As evidenced by the large and
ever growing number of papers on cross layer design over
the last few years, we expect that there will be no shortage
of cross layer ideas based on piecemeal approaches. The
growth of the “knowledge tree” on cross layer design has
been exponential. However, any piecemeal design jointly
over multiple layers does not bring more structured thinking process than the ad hoc design of just one layer. What
seems to be lacking is a level ground for fair comparison
among the variety of cross layer designs, a unified view
on how to and how not to layer, and fundamental limits
on the impacts of layer-crossing on network performance
and robustness metrics.
“Layering as optimization decomposition” provides a
candidate for such a unified framework. It attempts at
shrinking the “knowledge tree” on cross layer design
rather than growing it. It is important to note that “layering as optimization decomposition” is not the same as
the generic phrase of “cross-layer optimization”. What is
unique about this framework is that it views the network as
the optimizer itself, puts the end user application needs as
the optimization objective, provides the globally optimal
performance benchmark, and leads to a systematic design
of decomposed solutions to attain the benchmark. Not
all architectural principles may be readily quantified, but
“layering as optimization decomposition” provides one of
the few promising directions towards a mathematical theory of network architectures.
B. How to pick utility objective functions? How to guarantee QoS?
Answers: First of all, in reverse engineering, utility
functions are implicitly determined by the given protocols
already, and are to be discovered rather than designed. In
forward engineering, utility functions can be picked based
on any of the following four considerations:
• First, elasticity of application traffic can be modeled
through utility functions.
• Second, utility can be defined by human psychological and behavioral models such as Mean Opinion
Score in voice applications.
• Third, utility functions provide a metric to define optimality of resource allocation efficiency.
• Fourth, different shapes of utility functions lead to
optimal resource allocations that satisfy some definition of fairness (e.g., α-fair utilities parameterized by
α > 0: U (x) = (1 − α)−1 x1−α [47] leads to α-fair
resource allocation).
In general, depending on who is interested in the outcome of network design, there are two types of objective
functions: sum of utility functions by end users, which
can be functions of rate, reliability, delay, jitter, or power
level, and a network-wide cost function by network operators, which can be functions of congestion level, energy
efficiency, network lifetime, or collective estimation error.
Utility functions can be coupled across the users, and may
not have an additive structure (e.g., network lifetime).
Maximizing a weighted sum of all utility functions is
only one of the possible formulations. An alternative is
multi-objective optimization to characterize the Paretooptimal tradeoff between the user objective and operator
objective. Another set of formulations is game-theoretic
between users and operators, or among users or operators
themselves.
While utility models lead to objective functions, the
constraint set of a NUM formulation incorporates the following two types of constraints. First is the collection of
physical, technological, and economic restrictions in the
communication infrastructure. Second is the set of peruser, hard, inelastic QoS constraints that cannot be violated at the equilibrium. This is in contrast to the utility
objective functions, which may represent elastic QoS demands of the users.
C. What kind of NUM formulations are there?
Answers: The basic NUM problem is the following formulation [29], known as monotropic programming and
studied since 1960s. TCP variants have recently been reverse engineered to show that they are implicitly solving
this problem, where source rate vector x is the only set
of optimization variables, and routing matrix R and link
capacity vector c are both constants:
P
maximize
s Us (xs )
subject to Rx ¹ c.
(1)
Utility functions Us are often assumed to be smooth, increasing, concave, and dependent on local rate only, although recent investigations have removed some of these
assumptions for applications where they are invalid.
Many of the papers on “layering as optimization decomposition” are special cases of the following generic
problem [8], one of the possible formulations of a generalized NUM for the entire protocol stack:
P
P
maximize
s Us (xs , Pe,s ) +
j Vj (wj )
subject to Rx ≤ c(w, Pe ),
x ∈ C1 (Pe ), x ∈ C2 (F) or ∈ Π,
R ∈ R, F ∈ F, w ∈ W.
(2)
Here, xs denotes the rate for source s and wj denotes
the physical layer resource at network element j. The
utility functions Us and Vj may be any nonlinear, monotonic functions. R is the routing matrix, and c are the
logical link capacities as functions of both physical layer
resources w and the desired decoding error probabilities
Pe . The issue of signal interference and power control
can be captured in this functional dependency. The rates
must also be constrained by the interplay between channel decoding reliability and other error control mechanisms like ARQ. This constraint set is denoted as C1 (Pe ).
The issue of rate-reliability tradeoff and coding is captured in this constraint. The rates are further constrained
by the medium access success probability, represented by
the constraint set C2 (F) where F is the contention matrix, or the schedulability constraint set Π. The issue of
medium access control (either random access or scheduling) is captured in this constraint. The sets of possible
physical layer resource allocation schemes, of possible
scheduling or contention based medium access schemes,
and of single-path or multi-path routing schemes are represented by W, F, R, respectively. The optimization variables are x, w, Pe , R, F. Holding some of the variables
as constants and specifying some of these functional dependencies and constraint sets will then lead to a special
class of this generalized NUM formulation. As discussed
in the answer to the last question, utility functions and
constraint sets can be even richer.
A deterministic fluid model is used in the above formulations. Stochastic network utility maximization is an
active research area, as discussed later in this section.
Whether it is the basic, general, or stochastic NUM, there
are three separate steps in the process: first formulate a
specific NUM problem, then devise a modularized and
distributed solution following a particular decomposition,
and finally explore the space of alternative decompositions that provide a choice of layered protocol stack and
coupling across the layers.
There are many remaining challenges in formulating
NUM. For example, it is still difficult to fully incorporate
BGP for inter-AS routing in the generalized NUM framework. Similarly, an optimization-based, unifying view on
wireless ad hoc network routing is lacking. Much further work remains to be done to model utility functions in
specific applications, especially inelastic, real-time applications such as VoIP and streaming media [25]. In a more
refined physical/link layer model, the option of forwarding rather than re-encoding at intermediate nodes must be
considered, as well as retransmission through ARQ.
D. Isn’t it just all about dual decomposition?
Answers: The basic idea of decomposition is to divide the original large problem into smaller subproblems,
which are then coordinated by a master problem by means
of some kind of signalling. Most of the existing decomposition techniques can be classified into primal decomposition and dual decomposition methods 1 . The former is based on decomposing the original primal problem,
whereas the latter is based on decomposing the Lagrange
dual of the problem. Primal decomposition methods have
the interpretation that the master problem directly gives
each subproblem an amount of resources that it can use;
the role of the master problem is then to properly allocate
the existing resources. In dual decomposition methods,
the master problem sets the price for the resources to each
subproblem which has to decide the amount of resources
to be used depending on the price; the role of the master
problem is then to obtain the best pricing strategy. Primal decomposition and dual decomposition can in fact be
inter-changed by introducing auxiliary variables [51].
Almost all the papers in the vast, recent literature on
NUM use a standard dual-based distributed algorithm.
1
This is not to be confused with primal-dual interior-point algorithm,
or primal driven network control, or primal penalty function approach.
Contrary to the apparent impression that such a decomposition is the only possibility, there are in fact many alternatives to solve a given network utility problem in different
but all distributed manners [52], including multi-level and
partial decompositions. Each of the alternatives provides
a possibly different network architecture with different engineering implications.
Alternative horizontal decomposition (i.e., distributed
control across geographically disparate network elements)
has been studied in [52]. Recent results on alternative vertical decomposition (i.e., modularized control over multiple functional modules or layers) scatter in an increasingly
large research literature. For example, on the topic of joint
congestion control, routing, and scheduling, different decompositions have been obtained in [1], [5], [16], [41],
[57], on the topic of joint congestion control and random
access, different decompositions have been obtained in
[62], [39], and on the topic of rate control for network coding based multicast, different decompositions have been
obtained in [43], [64], [65], [2], [67], [7]. A systematic
treatise on this variety of vertical decompositions is an interesting research direction that will contribute to a rigorous understanding of the architectural choices of allocating functionalities to control modules.
Coupling for generalized NUM can happen not only in
constraints, but also in the objective function, where the
utility of source s, Us (xs , {xi }i∈I(s) ), depends on both
its local rate xs and the rates of a set of other sources
with indices in set I(s). If Us is an increasing function of {xi }i∈I(s) , this coupling models cooperation in a
clustered system, otherwise it models competition such as
power control in wireless network or spectrum management in DSL. Such coupling in the objective function can
be decoupled [58] by first introducing auxiliary optimization variables and consistency equality constraints, thus
shifting coupling in objective to coupling in constraints,
then introducing “consistency prices” to decouple the consistency constraints. These consistency prices are iteratively updated through local message passing.
E. How do you know which decomposition to pick?
Answers: Even a different representation of the same
primal problem may change the duality and decomposability structures even though it does not change the optimal solution. It remains an open issue how to systematically explore the space of alternative vertical and horizontal decompositions, and thus the space of alternative
network architectures, for a given set of requirements on,
e.g., rate of convergence, symmetry of computational load
distribution, and amount of explicit message passing.
An intellectually bold direction for future research is to
explore if both the enumeration and comparison of alternative decompositions, horizontally and vertically, can be
carried out systematically or even be automated.
To enumerate the set of possible decompositions, one
has to take into account that transformations of the problem (e.g., change of variable) may lead to new decomposability structure, or turn a seemingly non-decomposable
problem into a decomposable one [21]. This would open
the door to even more alternative decompositions, each
of which has a different engineering implication to distributed and modularized network architecture.
To compare alternative decompositions and the associated variety of distributed algorithms, the following metrics all need to be considered: speed of convergence,
the amount and symmetry of message passing for global
communication, the distribution of local computational
load, robustness to errors, failures, or network dynamics,
the possibility of efficient relaxations and simple heuristics, and the ability to remain evolvable as the application needs change over time. Some of these metrics do
not have any quantitative units of measurement, such as
evolvability. Some do not have a universally agreed upon
definition, such as the measure of how distributed an algorithm is. Some are difficult to analyze accurately, such
as the rate of convergence. Application contexts lead to a
prioritization of these possibly conflicting metrics, based
on which, the “best” decomposition can be chosen from
the range of alternatives.
Summarizing, there are three stages of conceptual understanding of an optimization/decomposition view of
network architectures:
• First, layered and distributed network architectures
can be rigorously understood as decompositions of
an underlying optimization problem.
• Second, there are in fact many alternatives of decompositions and therefore alternatives of network architectures. Furthermore, we can systematically explore
and compare such alternatives.
• Third, there may be a methodology to exhaustively
enumerate all alternatives, to quantify various comparison metrics, and even to determine a priori which
alternative is the best according to any given combination of comparison metrics.
Many issues in the third stage of the above list remain
open for future research.
F. What about those nonconvex optimization formulations?
Answers: Nonconvex optimization formulations of
generalized NUM may appear for four types of reasons.
First, non-concave utilities, such as sigmoidal utility. Second, non-convex constraint set, such as lower bounds on
SIR as a function of transmit power vector, in the lowSIR regime of interference-limited networks. Third, integer constraints, such as those in single-path routing protocols. Fourth, convex constraint sets that would require a
description length that grows exponentially with the number of variables, such as certain schedulability constraints
in multihop interference models. In general, nonconvex
optimization is difficult in theory and in practice.
For example, nonconvex optimization often has nonzero duality gaps. A non-zero duality gap means that
the standard dual-based distributed subgradient algorithm,
and in general dual decomposition approaches, may lead
to suboptimal and even infeasible primal solutions and
instability in cross layer interactions. This very difficult problem can be tackled through a combination of
well-established and more recent optimization techniques
(e.g., sum-of-squares programming [53] and geometricsignomial programming [9]). For example, there have
been three recent approaches to solve nonconcave utility
maximization over linear constraints:
• [36] proposes a distributed, suboptimal heuristic (for
sigmoidal utilities) called “self-regulating” heuristics, which is shown to avoid link congestion caused
by sigmoidal utilities.
• [22] determines optimality conditions for the dualbased distributed algorithm to converge globally (for
all nonlinear utilities). The engineering implication
is that appropriate overprovisioning of link capacities
will ensure global convergence of the dual-based distributed algorithm even when user utility functions
are nonconcave.
• [19] develops an efficient but centralized method to
compute the global optimum (for a wide class of utilities that can be transformed into polynomial utilities), using the sum-of-squares method.
There are at least three very different approaches to
tackle the difficult issue of nonconvexity:
Go around nonconvexity: discover a change of variable
that turns the seemingly nonconvex problem into a convex one, determine conditions under which the problem is
convex or the KKT point is unique, or make approximations to make the problem convex.
Go through nonconvexity: use successive convex relaxations (e.g., Sum-of-squares, Signomial programming),
utilize special structures in the problem (e.g., difference of
convex, generalized quasiconcavity), or leverage smarter
branch and bound methods.
Go above nonconvexity: observe that optimization
problem formulations are induced by some underlying as-
sumptions on what the architectures and protocols should
look like. By changing these assumptions, a different,
much easier-to-solve or easier-to-approximate NUM formulations may result. This is referred to as design for
optimizability [24], which concerns with redrawing architectures to make the resulting generalized NUM easier to
solve, rather than optimization that tries to solve a given,
possibly difficulty NUM problem.
G. Isn’t fluid model with infinite backlog too restrictive?
Answers: When sessions (i.e., flows, connections, endusers) arrive and depart, packets come in bursts, channels
vary over time, and topology is subject to change, new
formulations of stochastic NUM become necessary, presenting new challenges on stability and performance characterization. Most of the known results concern stochastic
stability and validity of the deterministic fluid model, with
little characterization on the distribution of utility or userperceived delay induced by the distributions of stochastic
models at various levels.
Session level. For Poisson arrivals of sessions with exponentially distributed file size, [3], [14], [46] showed
that, for certain classes of utility functions under the timescale separation assumption 2 , the stability region of the
basic NUM is the largest possible, which is the capacity region formed by the fixed link capacities in the deterministic NUM formulation. Then [40], [56] extended
this stochastic stability result to the case without the timescale separation assumption. Extensions have recently
been carried out to other models [57], [69], with fluid limits and diffusion approximations proposed as well [31],
[32]. Recent results have also established session-level
stochastic stability for any filesize distribution with finite
moments (e.g., see [11] and references therein).
Packet level. There have been three sets of results that
appeared over the last two years: translating on-off HTTP
session utility into transport layer TCP utility (mapping
from microscopic to macroscopic model) [4], showing
many-flow asymptotical validation of fluid model (justifying the transition from microscopic to macroscopic
model) [15], [54], and demonstrating convergence behavior for stochastic noisy feedback [72].
Channel level. Channel variations offer both the
challenge to prove stability/optimality for existing algorithms and the ability to do opportunistic transmission
and scheduling. For example, in [5], stability and optimality are established for dual algorithms under channellevel stochastic for any convex optimization where the
constraint set has the following structure: a subset of the
2
Here time-scale separation means that the resource allocation algorithm converges before the number of sessions changes.
variables lie in a polytope and other variables lie in a convex set that varies according to an irreducible, finite-state
Markov chain. “Layering as optimization decomposition”
type of algorithms that only require instantaneous knowledge of the current channel state (e.g., queue-lengths) remain stable and optimal (in the expected sense).
Topology level. Very little has been explored on this
topic, which is important for battery based or highly mobile wireless ad hoc networks.
H. Is anyone actually going to use this framework?
Answers: The application of “layering as optimization
decomposition” has been illustrated through many case
studies carried out by various research groups in the last
couple of years, generating considerable general insights
in addition to the specific cross-layer designs.
In the terminology of the standard seven layer reference
model, it is well-known that physical layer algorithms
try to solve the data transmission problem formulated by
Shannon: maximizing data rate subject to vanishing error
probability constraints. Recent progress have put protocols in layers 2-4 of the standard reference model on a
mathematical foundation as well:
• The congestion control functionality of TCP has
been reverse engineered to be implicitly solving the
basic NUM problem (1). While heterogeneous congestion control protocols do not solve an underlying
NUM problem, its equilibrium and dynamic properties can still be analyzed through a vector field representation and Poincare-Hopf index theorem [60],
[61], which show that sufficiently small “degree of
heterogeneity” implies global uniqueness and local
stability of network equilibrium.
• IGP of IP routing are known to be variants of shortest path routing solvers, and the policy-based routing
protocol in BGP has recently been modeled as the
solution to the Stable Path Problem [20].
• Scheduling based MAC protocols are known to be
solving variants of maximum weight matching, and
random access (contention based MAC) protocols
have recently been reverse engineered as a noncooperative selfish utility maximization game [38],
[59].
Following is a non-exhaustive list of some of the recent publications using “layering as optimization decomposition”, with references given in the bibliography 3 . In
all these cases, a NUM problem that is more complicated
than the basic NUM represents a more general networking
3
We apologize in advance for any references we may have missed
and would appreciate any information about other citations.
problem encompassing more than congestion control, and
some functions of the Lagrange dual variables act as the
“layering variables”.
• Jointly optimal congestion control and adaptive coding or power control
• Jointly optimal congestion and contention control
• Jointly optimal congestion control and scheduling
• Jointly optimal routing and scheduling
• Jointly optimal routing and power control
• Jointly optimal congestion control, routing, and
scheduling
• Jointly optimal routing, scheduling, and power control
• Jointly optimal routing, resource allocation, and
source coding
• TCP/IP interactions and jointly optimal congestion
control and routing
• Network lifetime maximization
Industry adoption of “layering as optimization decomposition” has already started. For example, insights from
reverse-engineering TCP has lead to an improved version
of TCP implemented over the last several years: FAST
(Fast AQM Scalable TCP) [17], [26]. Putting end-user
application utilities as the objective function has lead to
a new way to leverage innovations in the physical and
link layers beyond the standard metrics such as bit error
rate, e.g., in “FAST Copper” Project (here FAST stands
for Frequency, Amplitude, Space, Time) for an order-ofmagnitude boost to rates in fiber/DSL broadband access
systems [18].
I. Who cares about convergence at time infinity under a
weird stepsize?
Answers: Understanding practical stepsize choices’
impacts in an asynchronous environment, characterizing
transient behaviors of iterative algorithms, and tightly
bounding the rate of convergence are all important and
under-explored topics in this area.
For example, for certain applications, if the resource
allocation (e.g., window size, signal-to-interference-ratio)
for a user drops below a threshold during the transient, the
user may be disconnected. In such cases, the whole idea
of equilibrium becomes meaningless. Invariance during
transients, instead of convergence in the asymptote, becomes a more useful concept: how fast can the algorithm
gets close enough to the optimum and stay in that region? Usually the overall system performance derived out
of a modularized design determines “how close is close
enough” for each module’s transients.
Similarly, in the objective function, utility as a function
of delay, jitter, and even the entire time series of resource
allocation during the transients needs to be further investigated.
J. Why should network operators optimize performance
in the first place?
Answers: Indeed, optimality is not the key point. Optimization is used here primarily as a modeling language
and a starting point to develop and compare architectural
choices, rather than defining a particular point of operation at global optimum. Suboptimal, but simple (low
spatial-temporal complexity) algorithms can be used in
various modules (e.g., the scheduling module), and, as
long as the suboptimality gap is bounded and the network
architecture is “good”, then the “damage” from the suboptimal design in one layer can be contained at the systems
level [41]. Similarly, stochastic dynamics may also wash
away the corner cases and be beneficial to the average system performance, if the network architecture is appropriately designed. In such cases, it is interesting to study the
meaning of utility-suboptimality in terms of degradation
to fairness.
Protocols and layered architectures are not just for maximizing the efficiency of performance metrics, such as
throughput, latency, and distortion, but also robustness
metrics, such as evolvability, scalability, availability, and
manageability. Interactions among layers introduce the
risks of losing robustness against unforseen demands arising over time or significant growth over space. Despite
the importance in practical network operations, these network X-ities remain as important yet fuzzy notions, and a
quantified foundation for them is long overdue [12]. Intuitively, “design by decomposition” enhances scalability
and evolvability, but may present risks to manageability
such as diagnosability and optimizability. This is in part
because layering means that each layer is limited in what
it can do (optimization variables in a decomposed subproblem) and what it can observe (a subset of constant
parameters and variables in other decomposed subproblems). Quantifying network X-ities, and trading-off network X-ities with performance metrics, in layered protocol stack design is a long-term, challenging direction.
Carrying the intellectual thread from “forward engineering” (solve a given problem) to “reverse engineering” (find the problem being solved by a given protocol) one step further to “design for optimizability”, it may
be that the difficulty of solving a particular set of subproblems also illustrates that the given decomposition was
conducted possibly in a wrong way and suggests that better alternatives exist.
Under the metrics of deployment cost and operations
cost, the success of IP networks comes down to the scal-
ability and evolvability of TCP/IP and the way control
is distributed and modularized. A long-term goal of the
framework of “layering as optimization decomposition”
is to provide a simple, relevant abstraction of what makes
a network successful in this sense.
III. K EY M ESSAGES
In summary, “layering as optimization decomposition”
is a unifying framework for understanding and designing
distributed control and cross-layer resource allocation in
wired and wireless networks. It has been developed by
various research groups over the last several years, and
is now emerging to provide a mathematically rigorous
and practically relevant approach to quantify the risks and
opportunities of modifying existing layered network architecture. It shows that network protocols in layers 2,
3, and 4 can be reverse-engineered as implicitly solving
some optimization-theoretic or game-theoretic problems.
By distributively solving generalized NUM formulations
through decomposed subproblems, we can systematically
generate layered protocol stacks. There are many alternatives for both horizontal decomposition into disparate network elements and vertical decomposition into functional
modules (i.e., layers). While queuing delay or buffer occupancy is often used as the “layering price”, it may sometimes lead to unstable interactions. A variety of techniques to tackle coupling and nonconvexity issues have
become available.
Some of the key messages obtained from many case
studies are outlined below, and, as briefly discussed in
this tutorial paper, even more open problems and new directions present themselves in this emerging area of research.
• Protocols in layers 2,3,4 can be reverse engineered.
Reverse engineering in turn leads to better design in
a rigorous manner.
• There is a unifying approach to cross-layer design, as
summarized in Section I of this paper.
• Loose coupling through “layering price” can be optimal, and congestion price (or queuing delay, or
buffer occupancy) is often the right “layering price”
for stability and optimality, with important exceptions as well.
• There are many alternatives in decompositions, leading to different divisions of tasks across layers and
even different time-scales of interactions.
• Convexity of the generalized NUM is the key to devising a globally optimal solution.
• Decomposability of the generalized NUM is the key
to devising a distributed solution.
Architecture, rather than optimality, is the most important theme.
More than just an ensemble of specific cross-layer designs for existing protocol stacks, “layering as optimization decomposition” is a mentality that views networks as
optimizers, a common language that allows researchers to
quantitatively compare alternative network architectures,
and a suite of methodologies that facilitates a systematic
design approach for modularized network architectures.
•
ACKNOWLEDGEMENT
We would like to gratefully acknowledge the collaborations and interactions on this topic with many colleagues,
including Stephen Boyd, Ma’ayan Bresler, Lijun Chen,
Dah-Ming Chiu, Neil Gershenfeld, Andrea Goldsmith,
Prashanth Hande, Jiayue He, Sanjay Hegde, Maryam
Fazel, Bob Fry, Cheng Jin, Koushik Kar, Frank Kelly, P.
R. Kumar, Jang-Won Lee, Ruby Lee, Lun Li, Ying Li,
Xiaojun Lin, Jiaping Liu, Zhen Liu, Asuman Ozdaglar,
Daniel Palomar, Pablo Parrilo, Jennifer Rexford, Devavrat
Shah, Sanjay Shakkottai, Ness Shroff, R. Srikant, Chee
Wei Tan, Ao Tang, Jiantao Wang, Xin Wang, David Wei,
Bartek Wydrowski, Lin Xiao, Edmund Yeh, and Junshan
Zhang.
The work summarized in this paper has been in part
supported by NSF grants CCF-0440043, CCF-0448012,
CNS-0417607, CNS-0427677, CNS-0430487, CNS0519880, DARPA grant HR0011-06-1-0008, AFOSR
FA9550-06-1-0297, and Cisco grant GH072605.
R EFERENCES
[1] M. Andrews, “Joint optimization of scheduling and congestion
control in communication networks,” in Proc. of CISS, 2006.
[2] S. Bhadra, S. Shakkottai, and P. Gupta, “Min-cost selfish multicast
with network coding”, Proc. IEEE NetCod Workshop, April 2006.
[3] T. Bonald and L. Massoulie, “Impact of fairness on Internet performance”, Proc. ACM Sigmetrics, pp. 82-91, 2001.
[4] C. S. Chang and Z. Liu, “A bandwidth sharing theory for a large
number of HTTP-like connections,” IEEE/ACM Trans. on Networking, vol. 12, no. 5, Oct. 2004.
[5] L. Chen, S. H. Low, M. Chiang, and J. C. Doyle, “Optimal joint
congestion control, routing, and scheduling design for wireless ad
hoc networks,” Proc. IEEE INFOCOM, April 2006.
[6] L. Chen, S. H. Low, and J. C. Doyle, “Joint congestion control and
medium access control for wireless ad hoc networks,” Proc. IEEE
INFOCOM, March 2005.
[7] L. Chen, T. Ho, S. H. Low, M. Chiang, and J. C. Doyle, “Rate
control for multicasting with network coding”, Preprint, 2006.
[8] M. Chiang, “Balancing transport and physical layers in wireless
multihop networks: Jointly optimal congestion and power control”
IEEE J. Sel. Area Comm., vol. 23, no. 1, pp.104-116, Jan. 2005.
[9] M. Chiang, “Geometric programming for communication systems”, Trends and Foundations in Communications and Information Theory, vol. 2, no. 1, pp. 1-156, Aug. 2005.
[10] M. Chiang, S. H. Low, R. A. Calderbank, and J. C. Doyle, “Layering as optimization decomposition”, To appear in Proceedings
of IEEE, Jan. 2007. A shorter version appeared in Proc. Conf. Inform. Sciences and Systems, March 2006.
[11] M. Chiang, D. Shah, and A. Tang, “Stochastic stability of network utility maximization: General filesize distribution”, Proc.
Allerton Conference, Sept. 2006.
[12] M. Chiang and M. Yang, “Towars X-ities from a topological
point of view: Evolvability and scalability”, Proc. Allerton Conf.
Oct. 2004.
[13] R. L. Cruz and A. Santhanam: “Optimal routing, link Scheduling, and power control in multihop wireless networks,” Proc. IEEE
INFOCOM, April 2003.
[14] G. de Veciana, T. J. Lee, and T. Konstantopoulos, “Stability
and performance analysis of network supporting elastic services”,
IEEE/ACM Trans. on Networking, vol. 9, no. 1, pp. 2-14, Feb.
2001.
[15] S. Deb, S. Shakkottai, and R. Srikant, “Asymptotic behavior of
Internet congestion controllers in a many-flow regime”, Math. Operations Research, 2005.
[16] A. Eryilmaz and R. Srikant, “Joint congestion control, routing
and MAC for stability and fairness in wireless networks,” in IEEE
J. Sel. Area Comm., vol. 24, no. 8, Aug. 2006.
[17] TCP FAST Project netlab.caltech.edu.
[18] FAST Copper Project www.princeton.edu/fastcopper.
[19] M. Fazel and M. Chiang, “Nonconcave network utility maximization through sum of squares method”, Proc. IEEE CDC, Dec.
2005.
[20] T. G. Griffin, F. B. Shepherd, and G. Wilfong, “The stable path
problem and interdomain routing,” IEEE/ACM Trans. on Networking, vol. 10, no. 2, pp. 232-243, April 2002.
[21] P. Hande, S. Rangan, and M. Chiang, “Distributed power control for optimal QoS assignment in wireless data networks”, Proc.
IEEE INFOCOM, April 2006.
[22] P. Hande, S. Zhang, and M. Chiang, “Distributed rate allocation
for inelastic flows”, Submitted to IEEE/ACM Trans. on Networking, preliminary version in Proc. IEEE INFOCOM, March 2005.
[23] J. He, M. Chiang, and J. Rexford, “TCP/IP interaction based
on congestion prices: Stability and optimality”, Proc. IEEE ICC,
June 2006.
[24] J. He, J. Rexford, and M. Chiang, “Design principles of manageable networks”, Preprint, 2006.
[25] J. Huang, Z. Li, M. Chiang, and A. K. Katsaggelos, “Pricingbased rate control and joint packet scheduling for multi-user wireless upllink video streaming”, Proc. IEEE Packet Video Workshop,
April, 2006.
[26] C. Jin, D. X. Wei, and S. H. Low. TCP FAST: motivation, architecture, algorithms, performance. Proc. of IEEE INFOCOM,
March 2004.
[27] B. Johansson and M. Johansson, “Primal and dual approaches
to distributed cross-layer optimization,” in Proc. 16th IFAC World
Congress, Prague, Czech republic, 2005.
[28] K. Kar, S. Sarkar, and L. Tassiulas, “Achieving proportional fairness using local information in Aloha networks,” IEEE Trans. on
Auto. Control, vol. 49, no. 10, pp. 1858–1862, October 2004.
[29] F. P. Kelly, A. Maulloo, and D. Tan, “Rate control for communication networks: Shadow prices, proportional fairness and stability,” J. Op. Res. Soc., vol. 49, no. 3, pp.237–252, March 1998.
[30] F. P. Kelly and T. Voice, “Stability of end-to-end algorithms for
joint routing and rate control”, Computer Communications Review, vol, 35, no. 2, pp. 5-12, 2005.
[31] F. P. Kelly and R. J. Williams, “Fluid model for a network op-
erating under a fair bandwidth-sharing policy”, Annals of Applied
Probability, vol. 14, no. pp. 1055-1083, 2004.
[32] P. Key and L. Massoulie, “Fluid models of integrated traffic and
multipath routing”, To appear in Queuing Systems, 2006.
[33] M. Kodialam and T. Nandagopal, “Charaterizing achievable rates
in multi-hop wireless networks: The joint routing and scheduling
problem”, Proc. ACM Mobicom, Sept. 2003.
[34] S. Kunniyur and R. Srikant. End-to-end congestion control: utility functions, random losses and ECN marks. IEEE/ACM Trans.
on Networking, 2003.
[35] R. J. La and V. Anantharam, “Utility-based rate control in the
Internet for elastic traffic,” IEEE/ACM Trans. on Networking, vol.
10, no. 2, pp. 272-286, April 2002.
[36] J. W. Lee, R. R. Mazumdar, and N. Shroff, “Non-convex optimization and rate control for multi-class services in the Internet,”
Proc. IEEE INFOCOM, Hong Kong, China, March 2004.
[37] J. W. Lee, M. Chiang, and R. A. Calderbank, “Price-based distributed algorithm for optimal rate-reliability tradeoff in network
utility maximization”, To appear IEEE Journal of Selected Areas
in Communications, 2006.
[38] J. W. Lee, M. Chiang, and R. A. Calderbank, “Utility-optimal
medium access control: reverse and forward engineering,” Proc.
IEEE INFOCOM, April 2006.
[39] J. W. Lee, M. Chiang, and R. A. Calderbank, “Jointly optimal
congestion and contention control in wireless ad hoc networks”,
To appear in IEEE Communication Letters, 2006.
[40] X. Lin and N. B. Shroff, “On the stability region of congestion
control”,, Proc. Allerton Conference, 2004.
[41] X. Lin and N. Shroff, “Impact of imperfect scheduling in cross
layer design,” Proc. IEEE INFOCOM, March 2005.
[42] J. Liu, M. Chiang, and H. V. Poor, “Stability of distributed dual
algorithm for convex optimization”, Preprint, 2006.
[43] D. S. Lun, N. Ratnakar, M. Mdard, R. Koetter, D. R. Karger, T.
Ho and E. Ahmed, “Minimum-cost multicast over coded packet
networks”, IEEE Trans. Inform. Theory, 2005.
[44] S. H. Low, “A duality model of TCP and queue management
algorithms,” IEEE/ACM Trans. on Networking, vol. 11, no. 4, pp.
525–536, Aug. 2003.
[45] S. H. Low and D. E. Lapsley, “Optimization flow control, I: basic
algorithm and convergence,” IEEE/ACM Trans. on Networking,
vol. 7, no. 6, pp. 861–874, December 1999.
[46] L. Massoulie and J. W. Roberts, “Bandwidth sharing and admission control for elastic traffic”, Telecommunication Systems, vol.
15, pp. 185-201, March 2000.
[47] J. Mo and J. Walrand, “Fair end-to-end window-based congestion
control,” IEEE/ACM Trans. on Networking, vol. 8, no. 5, pp. 556–
567, Oct. 2000.
[48] H. Nama, M. Chiang, and N. Mandayam, “Utility lifetime tradeoff in self regulating wireless sensor networks: A cross-layer design approach”, Proc. IEEE ICC, June 2006.
[49] T. Nandagopal, T. Kim, X. Gao, and V. Bharghavan, “Achieving MAC layer fairness in wireless packet networks,” Proc. ACM
Mobicom, Boston, USA, Aug. 2000.
[50] M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing time varying wireless networks”, IEEE J. Sel.
Area Comm., vol. 23, no. 1, pp. 89-103, Jan. 2005.
[51] D. Palomar and M. Chiang, “A tutorial to decompositon methods
for network utility maximization”, IEEE J. Sel. Area Comm., vol.
24, no. 8, pp. 1439-1450, Aug. 2006.
[52] D. Palomar and M. Chiang, “Alternative decompositions for distributed maximization of network utility: Framework and applications”, Proc. IEEE INFOCOM, April 2006.
[53] P. A. Parrilo, “Semidefinite programming relaxations for semialgebraic problems,” Mathematical Programming Series B, vol. 96,
no. 2, pp. 293-320, 2003.
[54] S. Shakkottai and R. Srikant, “Mean FDE models for Internet
congestion control under a many-flows regime”, IEEE Trans. Inform. Theory, vol. 50, no. 6, pp. 1050-1072, June 2004.
[55] R. Srikant, The Mathematics of Internet Congestion Control,
Birkhauser 2004.
[56] R. Srikant, “On the positive recurrence of a Markov chain describing file arrivals and departures in a congestion-controlled network”, IEEE Computer Communications Workshop, 2004.
[57] A. L. Stolyar, “Maximizing queueing network utility subject to
statbility: greedy primal-dual algorithm”, Queueing Systems, vol.
50, no. 4, pp. 401-457, 2005.
[58] C. W. Tan, D. Palomar, and M. Chiang, “Decoupling coupled
network utility maximization”, Proc. IEEE ICASSP, May 2006.
[59] A. Tang, J. W. Lee, J. Huang, M. Chiang, and R. A. Calderbank,
“Reverse engineering MAC”, Proc. IEEE WiOpt, April 2006.
[60] A. Tang, J. Wang, S. H. Low, and M. Chiang, “Equilibrium of
heterogeneous congestion control protocols”, Proc. IEEE INFOCOM, March 2005.
[61] A. Tang, D. Wei, S. H. Low, and M. Chiang, “Heterogeneous
congestion control: Efficiency, fairness, and control”, Proc. IEEE
ICNP, Nov. 2006.
[62] X. Wang and K. Kar, “Cross-layer rate control for end-to-end
proportional fairness in wireless networks with random access,”
Proc. ACM Mobihoc, May, 2005.
[63] J. Wang, L. Li, S. H. Low, and J. C. Doyle. “Cross-layer optimization in TCP/IP networks,” IEEE/ACM Trans. on Networking,
vol. 13, no. 3, pp. 582-595, August 2005.
[64] Y. Wu and S.Y. Kung, “Distributed utility maximization for network coding based multicasting: A shortest path approach”, IEEE
Journal on Selected Areas in Communications, Aug. 2006.
[65] Y. Wu, M. Chiang, and S.Y. Kung, “Distributed utility maximization for network coding based multicasting: A critical cut
approach”, Proc. IEEE NetCod Workshop, April 2006.
[66] Y. Xi and E. Yeh, “Optimal distributed power control, routing,
and congestion control in wireless networks”, Submitted to IEEE
Trans. Inform. Theory, 2005.
[67] Y. Xi and E. M. Yeh, “Distributed algorithms for minimum cost
multicast with network coding”, Proc. IEEE NetCod Workshop,
April 2006.
[68] L. Xiao, M. Johansson, and S. Boyd, “Joint routing and resource
allocation via dual decomposition,” IEEE Trans. Comm., vol. 52,
no. 7, pp. 1136-1144, July 2004.
[69] H. Ye, “Stability of data networks under optimization-based
bandwidth allocation”, IEEE Trans. Auto. Control, vo.. 48, no. 7,
pp. 1238-1242, July 2003.
[70] W. Yu and J. Yuan, “Joint source coding, routing, and resource
allocation for wireless sensor networks,” Proc. IEEE ICC, May
2005.
[71] C. Yuen and P. Marbach, “Price-based rate control in random access networks,” IEEE/ACM Trans. on Networking, vol. 13, no. 5,
pp. 1027–1040, December 2005.
[72] J. Zhang, D. Zheng, and M. Chiang, “Impacts of stochastic noisy
feedback in network utility maximization,” Preprint, 2006.